Chapter 7: Capacitors

Table of Contents

7.1 Introduction to Capacitors

Capacitors are passive electrical components that store electrical energy in an electric field. They consist of two conductive plates separated by an insulator (dielectric) material. Capacitors are widely used in electronic circuits for various purposes, such as energy storage, filtering, and coupling or decoupling of signals.

An assorted collection of capacitors (though these are fairly large compared to the ones used in the modern technology we have all around us).

7.2 Capacitance

Capacitance is defined as the ratio of the electric charge stored in a device to the potential difference (voltage) across the device. Mathematically, this can be expressed as:

$C = \dfrac{Q}{V}$

where $C$ is the capacitance, $Q$ is the electric charge, and $V$ is the potential difference.

The unit of capacitance is the farad (F), which represents one coulomb of charge stored per volt of potential difference.

7.3 Types of Capacitors:

7.3.1 Parallel-Plate Capacitor:

The simplest and most common type of capacitor is the parallel-plate capacitor, which consists of two parallel conductive plates separated by a small distance. The capacitance of a parallel-plate capacitor is given by:

$C=\varepsilon_0 \dfrac{A}{d}$

where $\varepsilon_0$ is the vacuum permittivity $(8.854 \times 10^{-12} F/m)$ , $A$ is the area of the plates, and $d$ is the distance between the plates.

7.3.2 Spherical Capacitor

Another type of capacitor is the spherical capacitor. This type has an inner sphere surrounded concentrically by outer sphere. Capacitance for this type can be calculated as:

$C=\dfrac{1}{4 \pi \varepsilon_0} \biggl( \dfrac{r_b r_a}{r_b-r_a} \biggr) =\dfrac{1}{k} \biggl( \dfrac{r_b r_a}{r_b - r_a} \biggr)$

where $r_a$ is the radius of the inner sphere, $r_b$ is the radius of the outer sphere, and $k$ is the Coulomb constant equal to $\frac{1}{4 \pi \varepsilon_0}$ .

7.3.3 Cylindrical Capacitor

The cylindrical capacitor is yet another common type. This capacitor contains two cylinders, one inside the other, separated by a dielectric (non-conducting) medium. The formula for this type of capacitor is expressed as:

$C=L \dfrac{2 \pi \varepsilon_0}{\ln{\Bigl( \dfrac{r_b}{r_a} \Bigr) }}$

where $r_a$ and $r_b$ are, once again, the inner and outer radii, with $L$ being the length of the capacitor, and $\varepsilon_0$ the vacuum permittivity.

7.4 Energy Stored in a Capacitor

Sometimes it is necessary to calculate the actual energy stored in a capacitor. To do so, we can use any one of the following relations:

$U=\dfrac{Q^2}{2C}=\dfrac{CV^2}{2}=\dfrac{QV}{2}$

where $U$ is the total energy stored in the capacitor, $Q$ is the charge of the capacitor, $C$ is the capacitance, and $V$ is the potential difference (voltage) across the capacitor.

7.5 Capacitors in Series and Parallel

Capacitors can be included in a circuit in series and/or parallel configurations.

Circuits can be wired in series or parallel (or, most commonly, a combination of the two). More will be said on series and parallel soon. However, for the time being, think of series circuits as being one path from start to finish. On the other hand, parallel circuits contains several different branches through which current can flow.

7.5.1 Equivalent Capacitance (Series)

Capacitors aligned in series.

When two or more capacitors are place in series with one another, we can calculate an equivalent capacitance that treats the many separate capacitors as one capacitor. The equation to calculate equivalent capacitance is:

$C_{eq} = \dfrac{1}{\sum_i^n \dfrac{1}{C_i}}$

where $C_{eq}$ is the equivalent capacitors, and $\sum_i^n \frac{1}{C_i}$ is a simple sum of the reciprocal of all the capacitances. So, for a circuit with 3 capacitors in series:

$C_{eq} = \dfrac{1}{\dfrac{1}{C_1} + \dfrac{1}{C_2} + \dfrac{1}{C_3}}$

It is also important to note that each capacitor will carry the same charge for capacitors placed in series. In other words:

$Q_1 = Q_2 = Q_3 = ... = Q_n$

7.5.2 Equivalent Capacitance (Parallel)

Capacitors aligned in parallel.

For capacitors in parallel, determining the equivalent capacitance is very simple:

$C_{eq} = \sum_i^n C_i$

which is just the sum of all the capacitances in parallel.

Similarly, we can calculate the equivalent charge stored by these capacitors as:

$Q_{eq} = \sum_i^n Q_i$

which is just the sum of the charges stored on each capacitor in parallel.

Chapter Summary

In this chapter, we explored the fundamental concepts of capacitors, including their structure, capacitance, and energy storage capability. We discussed the parallel plate capacitor and derived the formula for its capacitance. Furthermore, we examined the charging and discharging processes of capacitors, as well as their behavior when connected in series or parallel. Capacitors are crucial components in various electronic circuits and understanding their properties is essential for analyzing and designing such circuits.

Continue to Chapter 8: Dielectrics

Are you enjoying this content? Read more from our Physics 2 course here!

Do you prefer video lectures over reading a webpage? Follow us on YouTube to stay updated with the latest video content!

Chapter 7: Capacitors

7.1 Introduction to Capacitors

7.2 Capacitance